Eight-dimensional Representation Research Articles

Data-Driven Radial Compressor Design Space Mapping

Abstract Estimates of turbomachinery performance trends inform system-level compromises during preliminary design. Existing empirical correlations for efficiency use limited experimental data, while analytical loss models require calibration to yield predictive results. From a set of 3708 radial compressor computations, this paper maps efficiency as a function of mean-line aerodynamics, and determines the governing loss mechanisms. An open-source turbomachinery design code creates annulus and blade geometry, then runs a Reynolds-averaged Navier–Stokes simulation for compressors sampled from the mean-line design space. Polynomial surface fits yield a continuous eight-dimensional representation of the design space for analysis, predicting efficiency with a root-mean-square error of 1.2% points. The results show a balance between surface dissipation in boundary layers and mixing loss due to casing separations sets optimum values for inlet Mach number, hub-to-tip ratio, de Haller number, and backsweep angle. Surface dissipation drives the effect of flow coefficient, with high surface areas at low values, and high velocities at high values. Compact compressor designs are achieved by increasing inlet Mach number, reducing hub-to-tip ratio, and minimizing the radial loading coefficient—all of which reduce efficiency approaching design space boundaries. An interactive web-based tool makes the results available to practising engineers, demonstrating large ensembles of automated designs and simulations as a higher-fidelity replacement for legacy empirical correlations in preliminary design.

Multiorder and structural mechanism of the LiRh<sub>2</sub>O<sub>4</sub> tetragonal phase formation

The results of the group-theoretical analysis of phase transitions in the LiRh2O4 are presented. It is found that the critical irreducible representation inducing phase transitions and multiorder in this substance is the eight-dimensional representation \(~{{k}_{{11}}}{{{\text{\tau }}}_{5}} + {{k}_{{10}}}{{{\text{\tau }}}_{2}}\). It is shown that the multiorder and the structural mechanism of formation of the tetragonal phase of lithium rhodonite are related to the displacements of oxygen atoms, the tilts of the [RhO]6 octahedra, and the ordering of the rhodium t2g orbitals.

A Formal Psychological Model of Classification Applied to Natural-Science Category Learning

The field of psychological science has seen major advances in the development of formal models of perceptual classification learning; however, little work has tested such models in real-world natural-category domains. In our current project, we aim to fill that gap by testing the ability of a formal exemplar model of classification to predict learning of rock categories in the geologic sciences. As a prerequisite for testing the model in this domain, we have conducted extensive work to derive a high-dimensional feature-space representation for the rock stimuli. An eight-dimensional representation yields good accounts of naive participants’ judgments of similarity among a large battery of rock-picture samples; furthermore, the eight dimensions have natural psychological interpretations. We then use the exemplar model in combination with the derived feature-space representation to successfully predict participants’ learning and generalization of a variety of scientifically defined rock categories. We discuss further steps for making use of the model and its associated feature-space representation to search for effective techniques of teaching categories in the science classroom.

Forced Symmetry-Breaking of Square Lattice Planforms

Equivariant bifurcation theory has been used to study pattern formation in various physical systems modelled by E(2)-equivariant partial differential equations. The existence of spatially doubly periodic solutions with respect to the square lattice has been the focus of much research. Previous studies have considered the four- and eight-dimensional representation of the square lattice, where the symmetry of the model is perfect. Here we consider the forced symmetry-breaking of the group orbits of translation free axial plan-forms in the four- and eight-dimensional representations. This problem is abstracted to the study of the action of the symmetry group of the perturbation on the group orbit of solutions. A partial classification for the behaviour of the group orbits is obtained, showing the existence of heteroclinic cycles and networks between equilibria. Possible areas of application are discussed including Faraday waves and Rayleigh-Benard convection. Subsequent studies will discuss other two- and three-dimensional lattices.

Computing the performance of unitary space-time group codes from their character table

Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment. Their success, however, depends on the design of cedes that achieve these promises. It is well known that unitary matrices can be used to design differentially modulated space-time codes. These codes have a particularly efficient description if they form a finite group under matrix multiplication. We show how to compute the parameters of such groups crucial for their use as space-time codes, using only the character table of the group. Since character tables for many groups are known and tabulated, this method could be used to quickly test, for a given group, which of its irreducible representations can be used to design good unitary space-time codes. We demonstrate our method by computing the eigenvalues of all the irreducible representations of the special linear group SL/sub 2/(F/sub q/) over a finite prime field F/sub q/ of odd characteristic, and study in detail the performance of a particular eight-dimensional representation of SL/sub 2/(F/sub 17/).

The braid group representations associated with some non-fundamental representations of Lie algebras

The braid group representations associated with the eight-dimensional representation of B3 and six-dimensional representation of A2, including the quantum Lie algebra connected with the latter, are explicitly calculated.

Eight-dimensional spinor representation of the Poincaré group

Finite-dimensional matrix representations of the Poincare group are discussed with particular emphasis on the eight-dimensional spinor representation. It is speculated that the complex eight-dimensional representation space might be interpreted as a more fundamental entity than Minkowski space, being in a sense a square root of the latter. One can model the usual position, momentum, and angular momentum variables of a particle of nonzero rest mass and arbitrary spin by real bilinear forms in the 8-spinor components, and obtain their correct equations of motion by subjecting the spinor to a Schrodinger-like evolution equation.

Algebraic Aspects of Trilinear Commutation Relations

An attempt is made to derive from the formalism of Schwinger's action principle, in a more convincing manner than previously described, a set of trilinear equal-time commutation relations which contains the commutation relations first discussed by H. S. Green as special cases. Matrix representations of field operators satisfying the trilinear commutation relations are considered. Two representations are explicitly discussed: a four-dimensional and an eight-dimensional representation. The representations considered and the bilinear equal-time commutation relations between the associated ``component fields'' obeying ordinary statistics are specified by irreducible representations of an algebra which is suggested by the trilinear commutation relations. The component fields associated with the same representation and of the same spin differ from each other in their bilinear equal-time commutation relations with other fields. This difference is reflected in the interactions into which the various fields can enter.

Bootstrap Mechanism and Symmetries of Strong Interactions

It is discussed what kinds of symmetries are consistent with the bootstrap mechanism in the baryon-pseucloscalar meson scattering problem. There, the one baryon exchange ap­ proximation is used. The masses of baryons ancl their coupling constants are calculated numerically. The global-, the cosmic- and the G 2-symmetrie;, are allowable, hut the 5'U(3) symmetry is excluded. ~ l. lntroduction Recently some authors 1 l have shown that the bootstrap mechanism predicts such a relation between the coupling constants of the strong interactions that is equal to the one of the octet scheme. The purpose of this article is twofold. We confirm Okabayashi's conjecture about the symmetry character imposed by the bootstrap mechanism on one hand, and on the other hand calculate the masses . of concerned bound states and coupling constants of strong interactions. As to the first problem, rr. Okabayashi 2 l pointed out that the bootstrap -mechanism introduces equalities between the coupling constants in the s-channel and the ones in the t (or u) -channel, and that the equalities give a kind of symmetry character of the strong interactions, which is not always represented by a group, in general. For instance, in Cutkosky's example, this symmetry character inevitably imposes the octet scheme, because he assumed a group character with the eight-dimensional representation and the tri-linear form of the interaction of the eight-component vector field beforehand. These assumptions are sufficient to restrict the symmetry to SU (3) among the groups which include the group of the charge independence. In other words, the equalities imposed by the bootstrap mechanism are accidentally fitted for the assumptions. In ~ 3, the conjecture mentioned above will be confirmed in the baryon-pseudoscalar meson scattering process, and it will be shown that the global symmetry, the cosmic symmetry, the G2 symmetry and the SU (3) symmetry accompanied with the R symmetry are consistent with the bootstrap mechanism, up to this step. The second problem is not so clear. Namely, there may not be any solu­ tion or there may be redundant solutions. Really, in the case of SU (3) sym­ metry there is an undesirable solution. As an example of the calculation of the bound states in a bootstrap mechanism, we have the calculation of the mass

SU 3 recoupling and fractional parentage in the 2s-1d shell

Explicit algebraic expressions have been calculated for both the SU 3 Wigner coefficients and the SU 3 Racah coefficients which are of particular interest in the recoupling problem involving 2s-1d shell nucleons and basis functions of SU 3 symmetry involving the SU 3 ⊃ SU 2 chain. The Wigner coefficients are those for the Kronecker products ( λ 1 μ 1)×( λ 2 μ 2), where ( λ 1 μ 1) is arbitrary and ( λ 2 μ 2) is the six-dimentional representation (20) of a single 2s-1d shell nucleon or one of the representations (02), (40), (21) for two coupled 2s-1d shell nucleons or the basic eight-dimensional representation (11). The racah coefficients are those involved in the coupling of two 2s-1d shell nucleons to a function of arbitrary (λμ). Calculations of SU 3 fractional parentage coefficients are illustrated by a few examples which have been evaluated without recourse to a full chain calculation for representations with large values of λ and μ. The SU 3 fractional parentage coefficients are used to give expressions for single-particle spectroscopic factors for 2s-1d shell nuclei.